TITLE: Derivative of $(a\,x)^{b\,x}$
QUESTION [5 upvotes]: is there any rule to differentiate the function $(a\,x)^{b\,x}$? 
I've got to find the derivative of $(x^2+1)^{\arctan x}$ and Wolfram|Alpha says the answer is 
$$\tan^{-1}(x) (x^2+1)^{\tan^{-1}(x)-1} \left(\frac{d}{dx}(x^2+1)\right)+\log(x^2+1) (x^2+1)^{\tan^{-1}(x)} \left(\frac{d}{dx}(\tan^{-1}(x))\right)$$
Is there any general rule to do that? 
Thanks.

REPLY [4 votes]: HINT $\rm\ \ (F^G)'\ =\ (e^{G\:ln\ F})'\: =\ F^G\ (GF'/F + G'\ ln\ F)$